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Calculus/calculus, Illumination problem

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Expert: Alon Mandes - 11/19/2008


Question

illumination diagram
Class: Calculus-1000

Consider a horizontal road illuminated by two lights, where Pi is the illumination power of Li, the light source and hi, are the corresponding heights of the two lamps.
Find the best possible lighting for the space found between the two lamps.

I was told the besxt equation to use would be
E=I(cos0)/r^2

I know r is the distance from the light source to the road. And I is the intensity of the light. and 0 is going to be the angle.

Answer

The drawing
We suppose that both light sources gives a a lightening brightness or clarity of E1=P1(cosθ1)/r1^2 & E2=P2(cosθ2)/r2^2. To find best possible lighting for the space found between the two lamps means
that E1+E2 are equal for every position in S. This demand means:
E1(θ1)+E2(θ2)=const -> E1(θ1)'+E2(θ2)'=0. Now our goal will be to
determine  θ1 & θ2 that satisfy our demand. & we will find θ1 &
θ2 by defining x (see the figure image attached) to a be a point in
the space s. Once we found x that satisfy the equation
E1(θ1)'+E2(θ2)'=0 , we will be able to determine θ1 & θ2. LEt's get
to work :
Cos(θ1)=x/r1 & r1^2=[x^2+H1^]^(1/2) -> E1(x)=P1*x*[H1^2+x^2]^(-3/2).
Cos(θ2)=(S-x)/r2 & r2^2=[(S-x)^2+H2^]^(1/2) ->
E2(x)=P2*(S-x)*[H2^2+(S-x)^2]^(-3/2).
Good , now let's derive E1 + E2 :
E1(x)+E2(x)=Const -> E1'(x)+E2'(x)=0 ->
E1(x)'=P1*[H1^2+x^2]^(-3/2)+P1*x*(-3/2)*(2x)*[H1^2+x^2]^(-5/2).
E1(x)'=P1*[H1^2+x^2]^(-3/2)-3P1*(x^2)*[H1^2+x^2]^(-5/2).
E2(x)'=-P2*[H1^2+(S-x)^2]^(-3/2)+P2*(S-x)*(-3/2)*
(-2)*(S-x)*[H2^2+(S-x)^2]^(-5/2).
E2(x)'=-P2*[H1^2+(S-x)^2]^(-3/2)+3P2(S-x)^2*[H2^2+(S-x)^2]^(-5/2).
Now we have to solve for x the equation :
P1*[H1^2+x^2]^(-3/2) - 3P1*(x^2)*[H1^2+x^2]^(-5/2) -
P2*[H1^2+(S-x)^2]^(-3/2) + 3P2(S-x)^2*[H2^2+(S-x)^2]^(-5/2) =0.
I'll leave it to you as an exercise.

Alon.

Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

Experience

1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

Organizations
Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

Education/Credentials
M.A in Mathematics & Bs.c in Electronics.

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